201002-A Short Note on Compressed Sensing

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1、AShortNoteonCompressedSensingwithPartiallyKnownSignalSupportTechnicalReport:TR-LJ-2009.02(UpdatedVersion)L.JacquesCommunicationsandRemoteSensingLaboratory(TELE)UniversitecatholiquedeLouvain(UCL)Louvain-la-Neuve,BelgiumFebruary21,2010AbstractThisshortnotestudiesavar

2、iationoftheCompressedSensingparadigmintroducedrecentlybyVaswanietal.,i.e.therecoveryofsparsesignalsfromacertainnumberoflinearmeasure-mentswhenthesignalsupportispartiallyknown.ThereconstructionmethodisbasedonaconvexminimizationprogramcoinedinnovativeBasisPursuitDeNois

3、e(oriBPDN).Underthecommon`2-delityconstraintmadeontheavailablemeasurements,thisoptimizationpromotesthe(`1)sparsityofthecandidatesignaloverthecomplementofthisknownpart.Inparticular,thispaperextendstheresultsofVaswanietal.tothecasesofcompressiblesignalsandnoisymeasure

4、ments.OurproofreliesonasmalladaptionoftheresultsofCandesin2008forcharacterizingthestabilityoftheBasisPursuitDeNoise(BPDN)program.WeemphasizealsoaninterestinglinkbetweenourmethodandtherecentworkofDavenportetal.onthe-stableembeddingsandthecancel-then-recoverstrategyap

5、pliedtoourproblem.Forbothapproaches,reconstructionsareindeedstabilizedwhenthesensingmatrixrespectstheRestrictedIsometryPropertyforthesamesparsityorder.Weconcludebysketchinganeasynumericalmethodrelyingonmonotoneoperatorsplittingandproximalmethodsthatiterativelysolvesi

6、BPDN.Keywords:SparseSignalRecovery,CompressedSensing,ConvexOptimization,InstanceOptimality.1IntroductionThetheoryofCompressedSensing(CS)[2,10]aimsatreconstructingsparseorcompressiblesignalsfromasmallnumberoflinearmeasurementscomparedtothedimensionalityofthesignalspac

7、e.Inshort,thesignalreconstructionispossibleiftheunderlyingsensingmatrixiswellbehaved,i.e.ifitrespectsaRestrictedIsometryProperty(RIP)sayingroughlythatanysmallsubsetofitscolumnsisclose"toanorthogonalbasis.Thesignalrecoveryisthenobtainedusingnon-lineartechniqueslaure

8、nt.jacques@uclouvain.be.ResearchsupportedbyBelgianNationalScienceFoundation(F.R.S.-FNRS).1basedonconvexoptimizationpromotingsignals